Optimization of Income

[superhume]

I've been having trouble with this question(i find the wording confusing). I'm unable to come up with an equation to determine the gross income.

"A television company agrees to establish a community antenna in a new suburb at a charge of $25 each for 1000 subscribers or less. To encourage more subscribers, it is agreed to give a discount of 10 cents to each subscriber for every additional 10 subscribers in excess of 1000. For the company to make the most income what number of subscribers should it have?"

Any help would be greatly appreciated!

 

[BigGlenntheHeavy]

\(\displaystyle Set \ up \ a \ table \ and \ you'll \ get \ f(x) \ = \ (25-.1x)(1000+10x) \ = \ (100+x)(250-x).\)

\(\displaystyle Hence \ f'(x) \ = \ (100+x)(-1)+(250-x)(1) \ = \ 150-2x \ = \ 0 \ \implies \ x \ = \ 75.\)

\(\displaystyle f(75) \ = \ (25-7.5)(1000+750) \ = \ 30,625 \ max \ the \ cable \ company \ can \ make \ by \ giving \ each\)

\(\displaystyle \ subscriber \ a \ discount \ of \ 7.50 \ when \ the \ subscribers \ reach \ a \ total \ of \ 1750.\)

\(\displaystyle Note: \ I \ edited \ this \ thread \ due \ to \ a \ misreading \ on \ my \ part.\)